Since the Higgs boson discovery more than two years ago, measurement of the Higgs couplings has been a major priority of the CERN Large Hadron Collider (LHC). Sensitivity to these couplings comes from measurements of Higgs production and decay event rates in particular production and decay modes.  However, the coupling extraction suffers from a "flat direction": if all the Higgs production couplings are increased by a common factor while the Higgs is also allowed to decay to a new, unobserved final state, the rates measured at the LHC would be unchanged.  In this talk I'll describe some recent attempts to get a handle on this scenario through studies of a concrete model (the Georgi-Machacek model, containing isospin-triplet scalars) in which the Higgs coupling to W or Z boson pairs is larger than in the Standard Model.